2nd Hour Pre-Calculus B (Spring 2012): April 2012

Tuesday, April 24, 2012

Properties of Exponents

$a^{x}\times a^{y}= a^{x+y}$

$\frac{a^{x}}{a^{y}}= a^{x-y}$

$\left ( a^{x} \right )^{y}= a^{xy}$

$a^{-x}= \frac{1}{a^{x}}$

$a^{0}= 1$

Exponential Functions

$y= a\cdot b^{x-c}+d$

Exponential Growth (b > 1)

Exponential Decay (0 < b < 1)

a : vertical stretch (changes y-int)

b : growth rate

c : horizontal shift

d : vertical shift

Compound Interest

$A= P\left ( 1+\frac{r}{n} \right )^{nt}$

A : present value

P : principal value

r : annual interest rate (APR)

n : number of compounding periods per year (1 month = 1/12 year)

t : time

Thursday, April 19, 2012

Law of Cosine and Heron's Formula

Law of Cosines are used to solve oblique triangles when you are given either three sides of the triangle (SSS), or if you are given two sides and their included angle (SAS).

This equation is best used when you are given all three sides of the triangle:

$\cos A= \frac{b^2 + c^2 - a^2}{2 b c}$

This equation is best used when you are given two sides and their included angle:

$a^2 = b^2 +c^2 - 2bc\cos A$

Heron's Area Formula is used to find the area of an oblique triangle:

$Area = \sqrt{s (s-a)(s-b)(s-c)}$

What is "s"? s is equal to the semi-perimeter, which is half of the triangles perimeter.

$s = \frac{a+b+c}{2}$

Wednesday, April 18, 2012

Oblique Triangles and Law of Sine

Oblique Triangles are any triangles that are not right traingles. They can be solved by the Law of Sines and Law of Cosines

Are of an Oblique Triangle

Example:

*Cross multiply and solve for a*

Tuesday, April 10, 2012

5.4 Sum and Difference Formulas

Sum and Difference Formulas are best used when an angle isn't obvious on the unit circle.

The two main purposes of these formulas are finding exact values of other trig functions and simplifying expressions to find other identities

Sum and Difference Formulas for sin, cos, and tan:

Example 1:

Find the exact value of cos 75°

cos75°=cos(30°+45°)

=cos 30° cos 45° - sin 30° sin 45°

You can check your answer by plugging it into your calculator. cos 75° is about 0.259, so our answer is correct!

Example 2: Find the exact value of

Double-Angle Identities

Sine:

Cosine:

There are three different versions of the cosine double angle identity, which can be found by deriving from each other.

Original Identity:

To derive a new identity substitute:

New versions of identity:

Tangent:

Thursday, April 5, 2012

Power-Reducing Formulas, Half-Angle Formulas, Sum-to-Product Formulas, Product-to-Sum Formulas

Power-Reducing Formulas

How to derive the $\sin ^2\Theta$ power-reducing formula:

Start out with an formula that includes both cosine and sine

$\cos 2\Theta = 1-2sin^2\Theta$
Solve for $\sin ^2\Theta$

First add $2\sin ^2\Theta$ to each side. This will cancel it on the right side of the formula. Then you must subtract $\cos2\Theta$ from each side. Your formula should now look like this:

$2\sin ^2\Theta = 1-cos2\Theta$

Now divide each side by 2 and you will get the sine power-reducing formula.

Half-Angle Formulas

How to derive the sine half-angle formula:

$\sin ^2\Theta =\frac{1-\cos 2\Theta }{2}$
$u=2\Theta$
This allows you to substitute u into the equation for theta.

Now your formula should look like:

$\sin ^2\frac{u}{2}=\frac{1-\cos u}{2}$
Take the square root of each side. Remember that the square root of the right side will be $\pm$
The formula should now look like this:

$sin\frac{u}{2}=\pm \sqrt{\frac{1-\cos u}{2}}$
Your final answer can either be + or - but it can not be both! The sign of the final answer depends where it is located on the unit circle.

If you wanted to find $\sin \frac{\pi }{8}$ you would have to use $\frac{\pi }{4}$ in the half-angle formula.

Sum-to-Product Formulas

Product-to-Sum Formulas